1. Field of the Invention
The invention generally relates to the field of well management and petroleum reservoir simulation. More specifically, this invention generally relates to the determination of an equivalent well block radius that takes into consideration the effects of three dimensional flow.
2. Description of Related Art
In the petroleum industry, well management techniques, such as reservoir simulation, are used to predict the future performance of a reservoir. Generally, well management techniques allow reservoir simulation engineers to research, investigate, analyze, and evaluate different field-development schemes, hydrocarbon-depletion strategies, pressure-maintenance requirements, and drilling and workover rig time estimates. Such research, investigation, analysis, and evaluation assists well managers in developing producing strategies. A typically producing strategy considers when and how to initiate actions such as workovers, shut-ins, drilling, connecting a well to a surface facility, or abandoning a well.
Since the development of early parallel computers, the oil industry has focused on the attractive speed of such machines to solve complicated well management problems. Initial inquiries and research focused on the preliminary question of whether it would even be possible to develop a truly parallel reservoir simulator. Contrary to seismic imaging algorithms and processes, it is well known by those skilled in the art that petroleum reservoir simulator algorithms are not naturally parallel as they are more recursive in nature and variables in such simulators are strongly interdependent on each other. Particularly, variables tend to display strong coupling and nonlinearity. If parallel code could be developed, the speed of computations in a reservoir simulator would increase by at least an order of magnitude, and as a result, reservoir simulators could handle larger and more complex scenarios. Beneficially, parallel code was subsequently developed for reservoir simulators that ultimately aided in the understanding of fluid flow in a complex reservoir. Moreover, parallelization of reservoir simulators provided an avenue to properly model and understand reservoir heterogeneities which resulted ultimately in more accurate well management predictions.
Parallelized reservoir simulators typically divide a simulated reservoir field into a series of blocks or grid cells. For example, a grid size of fifty (50) meters or less is often used for small and medium-size reservoir simulations. By contrast, simulations of many of the world's largest reservoirs, such as the giant reservoirs in the Middle East, use grid block sizes of 250 meters or larger. Even with such large grid block sizes, a model of such giant reservoirs can easily result in a need to process and analyze more than one (1) million grid block cells. Furthermore, each of these one (1) million or more cells can be perforated at the cell boundaries thereby resulting in three dimensional flow (i.e., both horizontal and vertical flow) of petroleum through and across one or more of the grid cells. Particularly, it is well known that for partially completed wells the bottom perforations are subject to strong vertical flow. Accordingly well prediction measurements and analysis should take into account vertical flow as well as horizontal flow in order to increase the accuracy of such well predictions in a parallel reservoir simulator.
As an example, water and gas coning is frequently encountered in oil wells. Gas coning is the tendency of gas in a gas-drive reservoir to push oil downward in an inverse code contour toward casing perforations. In extreme conditions, gas, not oil, will be produced from the well. Water coning is the change in the oil-water contact profiles as a result of drawdown pressures during production. Coning often occurs in vertical or slightly deviated wells and is affected by the characteristics of the fluids involved and the ratio of horizontal to vertical permeability. For such wells, rate calculations by a reservoir simulator are critical in order to design field production rates at certain levels to avoid bottom water coning or gas coning from the gas cap. A determined equivalent well block radius is used to calculate well productivity indices and ultimately production and injection rate of a well, as is known and understood by those skilled in the art. Any errors or inaccuracies in the calculation of the equivalent well block radius will yield erroneous well productivity indices and, accordingly, erroneous well rates. Ultimately, an erroneous equivalent well block radius calculation can lead to inaccurate predictions of water and gas breakthrough times at the wells.
The conventional approach to calculate equivalent well block radiuses for grid block cells in a reservoir simulator, well productivity indices for each well grid block cell, and ultimately production and injection rate of a well, as known and understood by those skilled in the art, only produces accurate results for two dimensional, horizontal well flow. The conventional approach does not take into account flow in the vertical direction. Therefore, for any well where vertical flow is an important factor, the conventional approach will yield erroneous well production or injection rates. For simulations of giant reservoirs, such as those in the Middle East, inaccurate calculations of well block radiuses for over 100 grid block cells can yield disastrously inaccurate results.
There exists in the art a numerical solution, in the form of a software package, for the productivity index and well production rate of an arbitrarily shaped well that does account for three dimensional, or horizontal and vertical, flow. Nevertheless, no specific formula is provided in the art to determine the productivity indices for a well with three dimensional flow. Accordingly, the numerical solution is difficult to calculate and cannot be coded into a reservoir simulator such that it can be easily executed in parallel processes. Furthermore, the existing numerical approach is computationally expensive and difficult to implement in legacy simulators that rely on analytical equations rather than numerical methods.